Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Backward Stochastic Variational Inequalities with Locally Bounded Generators

Backward Stochastic Variational Inequalities with Locally Bounded Generators The paper deals with the existence and uniqueness of the solution of the backward stochastic variational inequality: -dYt + (Yt ) dt F (t, Yt , Zt ) dt - Zt dBt , 0 t < T YT = , where F satisfies a local boundedness condition. Mathematics Subject Classification 2010: 60H10, 93E03, 47J20, 49J40. Key wor: backward stochastic differential equations, subdifferential operators, stochastic variational inequalities. 1. Introduction We consider the following backward stochastic variational inequality (BSVI): (1) -dYt + (Yt ) dt F (t, Yt , Zt ) dt - Zt dBt , 0 t < T YT = , where {Bt : t 0} is a standard Brownian motion, is the subdifferential of a convex l.s. function , and T > 0 is a fixed deterministic time. The study of the backward stochastic differential equations (BSDE) (equation of type (1) withouhe subdifferential operator) was initiated * The work was supported by IDEAS project, no. DRU/89/1.5/S/49944 project. 241/05.10.2011 and by POS- ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU by Pardoux and Peng in [11] (see also [12]) where is proved the existence and the uniqueness of the solution for the BSDE under the assumption of Lipschitz continuity of F with respeco y and z and square integrability of and F (t, 0, 0). The more general case of scalar BSDE with one-sided reflection and associated optimal control problems was considered by El Karoui et al. [8] and with two-sided reflection associated with stochastic game problem by Cvitanic, Karatzas [6] (see also [3] and [7] for the investigation of zero-sum two-player stochastic differential games whose cost functionals are given by controlled reflected BSDE). On the other hand, it is worth to mention the backward in time problems in mechanics of continua, since a large number of physical phenomena lea to these new non-standard problems. Specify that for improperly posed problems the solutions will not exist for arbitrary data and not depend continuously on the data (see e.g. [5], [4] and references therein). s The standard work on BSVI is that of Pardoux and Ra¸canu [13], which give a proof of existence and uniqueness of the solution for (1) under the following assumptions on F : monotonicity with respeco y (in the sense that y - y, F (t, y , z) - F (t, y, z) |y - y|2 ), Lipschitzianity with respeco z and a sublinear growth for F (t, y, 0), |F (t, y, 0)| t + L |y| , (t, y) [0, T ] × Rm. It is proved thahere exists a unique triple (Y, Z, K) such that Yt +KT -Kt =+ F (s, Ys , Zs ) - , with dKt (Yt ) dt. Moreover the process K is absolute continuous with respeco dt. In [14] the same authors extend the results from [13] to the Hilbert spaces framework. Using a mixed Euler-Yosida scheme, Maticiuc, Rotenstein provided in [9] numerical results concerning the multi-valued stochastic differential equation (1). Our paper generalize the previous existence and uniqueness results for (1) by assuming a local boundedness condition (instead of sublinear growth of F ), i.e. # F (s) # where F (t) = sup |F (t, y, 0)| . |y| Concerning to this requirement on F we remark that a similar one was considered by Pardoux in [10] for the study of BSDE. More precisely, his BSVIs WITH LOCALLY BOUNDED GENERATORS result is the following: if L2 (; Rm ), F (t, 0, 0) L2 ( × [0, T ] ; Rm ), F is monotone with respeco y, Lipschitz with respeco z and there exists a deterministic continuous increasing function such that (t, y) [0, T ] × Rm , |F (t, y, 0)| |F (t, 0, 0)| + (|y|) , P-a.s, then there exist a unique solution for BSDE (1) with 0. This result was generalized by Briand et al. [2]. The article is organized as follows: in the next Section we prove some a priori estimates and the uniqueness result for the solution of BSVI (1). Section 3 is concerned on the existence result under two alternative assumptions (which allow to obtain the absolute continuity of the process K) and Section 4 establishes the general existence result. In the Appendix we presents, following [15], some results useful throughouhe paper. 2. Preliminaries; a priori estimates and the uniqueness result Let {Bt : t 0} be a k-dimensional standard Brownian motion ined on some complete probability space (, F, P). We denote by {Ft : t 0} the natural filtration generated by {Bt : t 0} and augmented by N , the set of P- null events of F, Ft = {Br : 0 r t} N . We suppose thahe following assumptions hol: (A1 ) : Rm is a FT -measurable random vector, (A2 ) F : × [0, T ] × Rm × Rm×k Rm satisfies that, for all y Rm , z Rm×k , (, t) - F (·, ·, y, z) : × [0, T ] Rm is progressively measurable stochastic process, and there exist µ : × [0, T ] R and : × [0, T ] R+ progressively measurable stochastic processes T with 0 |µt | + 2 dt such that, for all t [0, T ], y, y Rm t and z, z Rm×k , P-a.s.: (Cy ) y - F (t, y, z) : Rm Rm is continuous, (My ) y - y, F (t, y , z) - F (t, y, z) µt |y - y|2 , (Lz ) |F (t, y, z ) - F (t, y, z)| t |z - z|, (By ) # F (s) 0, # where, for 0, F (t) = sup |F (t, y, 0)| , |y| (A3 ) : Rm (-, +] is a proper, convex l.s. function. ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU The subdifferential of is given by (y) = {^ Rm : y , v-y +(y) y ^ (v), v Rm }. We ine Dom() = {y Rm : (y) < }, Dom() = {y Rm : (y) = } Dom() and by (y, y ) we understand that y Dom () ^ and y (y). ^ Recall that Dom() = Dom(), Int(Dom()) = Int(Dom()). Let > 0 and the Moreau-Yosida regularization of : (y) = inf (2) = 1 |y - v|2 + (v) : v Rm 2 1 |y - J (y)|2 + (J (y)) , 2 where J (y) = (Im×m + )-1 (y). Remark that is a C 1 convex function and J is a 1-Lipschitz function. s We mention some properties (see Br´zis [1], and Pardoux, Ra¸canu e [13] for the last one): for all x, y Rm (a) (y) = (y) = (3) y - J (y) (J y), 1 (b) | (x) - (y)| |x - y| , (c) (x) - (y), x - y 0, (d) (x) - (y), x - y -( + ) (x), (y) . p We denote by Sm [0, T ] the space of (equivalent classes of) progressively measurable and continuous stochastic processes X : × [0, T ] Rm such that E supt[0,T ] |Xt |p if p > 0, and by p (0, T ) the space of (equivalent m classes of) progressively measurable stochastic process X : × [0, T ] Rm such tha |Xt |2 dt P-a.s. , if p = 0, T p/2 |Xt | dt if p > 0. For a function g : [0, T ] Rm , let us denote by g g on [0, T ] i.e. g T n-1 the total variation of = sup i=0 |g (ti+1 ) - g (ti )| : n N , 0 = t0 < t1 < · · · tn = T BSVIs WITH LOCALLY BOUNDED GENERATORS and by BV ([0, T ] ; Rm ) the space of the functions g : [0, T ] Rm such that g T < (BV ([0, T ] ; Rm ) equipped with the norm ||g||BV ([0,T ];Rm ) = |g(0)| + g T is a Banach space). 0 inition 1. A pair (Y, Z) Sm [0, T ] × 0 m×k (0, T ) of stochastic processes is a solution of backward stochastic variational inequality (1) if 0 there exists K Sm [0, T ] with K0 = 0, such tha (a) (b) 0 s | (Yt )| dt + |F (t, Yt , Zt )| dt a.s., s s dKt (Yt ) dt, a.s. that is: P-a.s., y(r) - Yr , dKr + (Yr )dr (y(r))dr, y C([0, T ]; Rd ), 0 t s T, and, P-a.s., for all t [0, T ] : (4) Yt + KT - Kt = + F (s, Ys , Zs ) - Zs dBs (we also say thariplet (Y, Z, K) is solution of equation (1)). Remark 2. If K is absolute continuous with respeco dt, i.e. there exists a progressively measurable stochastic process U such that |Ut | dt a.s. and Kt = Us , for all t [0, T ] , then dKt (Yt ) dt means Ut (Yt ) , dt-a.e., a.s. ~ ~ If dKt (Yt ) dt and dKt (Yt )dhen we clearly have | (Yt )| dt + ~ |(Yt )|dt a.s. and, using the subdifferential inequalities s t ~ Yr - Yr , dKr + (Yr )dr s t ~ (Yr )dr, (Yr )dr, ~ ~ Yr - Yr , dKr + ~ (Yr )dr ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU we infer that, for all 0 t s T (5) ~ ~ Yr - Yr , dKr - dKr 0, a.s. Let a, p > 1 and (6) Vt = Vt = 0 t µs + a 2 , 2np s where np = (p - 1) 1. Denote 1 Sm + ,p [0, T ] = 0 Y Sm [0, T ] : a > 1, E sup |eVs Ys |p < . Remark that if µs and 2 are deterministic functions then, for all p > 1, s 1+ p Sm ,p [0, T ] = Sm [0, T ]. Proposition 3. Let (u0 , u0 ) and assumptions (A1 -A3 ) be satis^ fied. Then for every a, p > 1 there exists a constant C such that for every (Y, Z) solution of BSDE (1) satisfying E sup epVs |Ys - u0 |p + E 0 T u eVs (|^0 | + |F (s, u0 , 0)|) the following inequality hol P-a.s., for all t [0, T ] : E Ft sup e s[t,T ] Vs (Ys - u0 ) + p/2 |Zs | p/2 Ft (7) Ft F e |(Ys ) - (u0 )| epVs |Ys - u0 |p-2 1Ys =u0 |Zs |2 epVs |Ys - u0 |p-2 1Ys =u0 |(Ys ) - (u0 )| C EFt epVT | - u0 |p T p u eVs |^0 | eVs BSVIs WITH LOCALLY BOUNDED GENERATORS and, for every R0 > 0 and p 2, p/2 R0 EF p/2 |F (s, Ys , Zs )| Ft (8) epVs |Ys - u0 |p-2 1Ys =u0 |F (s, Ys , Zs )| p/2 p/2 C EFt epVT | - u0 |p + R0 ·E F 1p2 # Fu0 ,R0 (s) + + R0 s Ft where # eVs Fu0 ,R0 (s) + 2R0 |s | # Fu0 ,R0 (t) = sup |y-u0 |R0 |F (t, y, 0)| . Proof. We can write Yt - u0 = - u0 + [F (s, Ys , Zs ) - dKs ] - Zs dBs . Let R0 0. The monotonicity property of F implies that, for all |v| 1 : F (t, u0 + R0 v, z) - F (t, y, z) , u0 + R0 v - y µt |u0 + R0 v - y|2 , and, consequently R0 F (t, y, z) , -v + F (t, y, z) , y - u0 µt |u0 + R0 v - y|2 + |F (t, u0 + R0 v, z)| |y - R0 v - u0 | # µt |u0 + R0 v - y|2 + Fu0 ,R0 (t) + t |z| |y - R0 v - u0 | # µt |u0 + R0 v - y|2 + Fu0 ,R0 (t) |y - R0 v - u0 | np 2 a 2 # + |z| Fu0 ,R0 (t) (|y - u0 | + R0 ) t |y - R0 v - u0 |2 + 2np 2a np # 2 2 + + t |y - u0 |2 -2R0 v, y - u0 +R0 |v|2 + |z|2 R0 Fu0 ,R0 (t) + R0 t 2a np 2 # |z| . + Fu0 ,R0 (t) + 2R0 |t | |y - u0 | + t |y - u0 |2 + 2a ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU Taking sup|v|1 , we have R0 |F (t, Yt , Zt )| dt + Yt - u0 , F (t, Yt , Zr ) dt # # 2 + R0 Fu0 ,R0 (t) + R0 t + Fu0 ,R0 (t) + 2R0 |t | |Yt - u0 | np |Zt |2 . + |Yt - u0 |2 dVt + 2a From the subdifferential inequalities we have |(t, Yt )-(t, u0 )| [(t, Yt )- (t, u0 )]+2|^0 ||Yt -u0 | and [(t, Yt )-(t, u0 )]dt Yt -u0 , dKt . Therefore u |(t, Yt ) - (t, u0 )|dt Yt - u0 , dKt + 2|^0 ||Yt - u0 |dt. From the above it u follows that (9) [R0 |F (t, Yt , Zt )| + |(Yt )- (u0 )|] dt+ Yt -u0 , F (t, Yt , Zt ) dt-dKt # # 2 + R0 Fu0 ,R0 (t) +R0 t dt+ Fu0 ,R0 (t) +2R0 |t | +2 |^0 | |Yt -u0 | dt u np 2 2 |Zt | . + |Yt - u0 | dVt + 2a For R0 = 0, inequality (7) clearly follows from (9) applying Proposition 11 from Appendix. For R0 > 0 we moreover deduce, using once again Proposition 11, inequality (8). Remark 4. Denoting = eVT | - u0 | + 0 0 eVs |^0 | + u eVs we deduce that, for all t [0, T ] : (10) 1/p |Yt | |u0 | + C e-Vt EFt p 1/p , a.s. Corollary 5. Let p 2. We suppose moreover thahere exist r0 , c0 > 0 such that #0 ,r0 = sup { (u0 + r0 v) : |v| 1} c0 . u Then r0 EFt (11) C E + t Ft p/2 p/2 e d K s p pVT | - u0 | + #0 ,r0 u p/2 - (u0 ) t p eVs |^0 | u eVs BSVIs WITH LOCALLY BOUNDED GENERATORS Proof. Let an arbitrary function v C ([0, T ] ; Rm ) such that v T 1. From the subdifferential inequality u0 + r0 v (t) - Yt , dKt + (Yt )dt (u0 + r0 v (t)) dt, we deduce that r0 d K t + (Yt )dt Yt - u0 , dKt + #0 ,r0 dt. Since Yt - u0 , u0 + (u0 ) (Yt ), then ^ u r0 d K Therefore r0 d K t + Yt - u0 , F (t, Yt , Zt ) dt - dKt #0 ,r0 - (u0 ) dt + |Yt - u0 | (|^0 | + |F (t, u0 , 0)|) dt u u np 2 2 |Zt | dt. + |Yt - u0 | dVt + 2a The inequality (11) follows using Proposition 11. Proposition 6 (Uniqueness). Let assumptions (A1 -A3 ) be satisfied. 0 ~ ~ Let a, p > 1. If (Y, Z) , (Y , Z) Sm [0, T ] × 0 m×k (0, T ) are two solutions of BSDE (1) corresponding respectively to and such that E sup epVs |Ys - ~ ~ ~s |p then for all t [0, T ] , epVt |Ys - Ys |p EFt epVT | - |p , P-a.s. ~ Y and there exists a constant C such that P-a.s., for all t [0, T ] : EFt ~ sup epVs |Ys - Ys |p + s[t,T ] Yt - u0 , dKt + |^0 | |Yt - u0 | dt + #0 ,r0 - (u0 ) dt. u u (12) ~ e |Zs - Zs |2 p/2 ~ C EFt epVT | - |p . + ,p 1 Moreover, the uniqueness of solution (Y, Z) of BSDE (1) hol in Sm 0 m×k (0, T ). [0, T ]× 0 ~ ~ Proof. Let (Y, Z), (Y , Z) Sm [0, T ] × 0 m×k (0, T ) be two solutions corresponding to and respectively. Then there exists p > 1 such that ~ p ~ ~ ~ Y, Y Sm [0, T ] and Yt - Yt = - + t dLs - t (Zs - Zs )dBs where ~ t ~ ~ ~ Lt = 0 [(F (s, Ys , Zs ) - F (s, Ys , Zs )) - (dKs - dKs )]. Since by (5) Ys - ~ ~ Ys , dKs - dKs 0, then, for all a > 1, ~ ~ ~ ~ Yt - Yt , dLt |Yt - Yt |2 µt dt + |Yt - Yt ||Zt - Zt |t dt np a 2 ~ ~ dt + |Zt - Zt |2 dt. |Yt - Yt |2 µt + 2np t 2a By Proposition 11, from Appendix, inequality (12) follows. ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU 1+ ~ ~ Let now p > 1 be such that (Y, Z) , (Y , Z) Sm ,p [0, T ] × 0 m×k (0, T ) are two solutions of BSDE (1) corresponding respectively to and . From ~ 1+,p the inition of space Sm [0, T ] there exists a > 1 such that ~ E sup |eVt Yt |p E sup |eVt Yt |p < . t[0,T ] t[0,T ] Consequently estimate (12) follows and uniqueness too. 3. BSVI - an existence result Using Proposition 3 we can prove now the existence of a triple (Y, Z, K) which is a solution, in the sense of inition 1, for BSVI (1). In order to obtain the absolute continuity with respeco dt for the process K it is necessary to impose a supplementary assumption. Let (u0 , u0 ) be fixed and ^ = C e2p T T V | - u0 |p + 0 p |^0 | u (13) where a, p > 1, C is the constant given by Proposition 3 and Vt is ined by (6). T If there exists a constant M such that || + 0 M, a.s. p p p u then T C e2p V T [(M + |u0 |) + |^0 | T ] and by (10) |Yt | |u0 | + EFt T 1/p 1/p |u0 | + C e2 V [M + |u0 | + |^0 | T ] , a.s. u We will make the following assumptions: (A4 ) There exist p 2, a positive stochastic process L1 ( × (0, T )), a positive function b L1 (0, T ) and a real number 0, such that (i) E+ () (ii) for all (u, u) and z Rm×k : ^ 1 u u, F (t, u, z) |^|2 + t + b (t) |u|p + |z|2 ^ 2 dP dt-a.e., (, t) × [0, T ] , and BSVIs WITH LOCALLY BOUNDED GENERATORS (A5 ) There exist M, L > 0 and (u0 , u0 ) such that: ^ (i) (ii) E+ () t L, a.e., t [0, T ] , (iii) || + M, a.s., , 1/p V (iv) R0 |u0 | + C e2 [M + |u0 | + |^0 | T ] u such that E # FR0 (s) < . We note that, if u, F (t, u, z) 0, for all (u, u) , then condition ^ ^ (A4 -ii) is satisfied with t = b (t) = = 0. For example, if = ID (the ¯ ¯ convex indicator of closed convex set D) and ny denotes the unit outward ¯ ¯ normal vector to D at y Bd D , then condition ny , F (t, y, z) 0 for ¯ yiel (A4 -ii) with t = b (t) = = 0. In this last case the all y Bd D It^'s formula for (y) = [distD (y)]2 and the uniqueness yiel K = 0. o ¯ We also remark that if F (t, y, z) = F (y, z) then assumptions (A5 ) becomes || + E+ () M, a.s., . Theorem 7 (Existence). Let p 2 and assumptions (A1 -A3 ) be satisfied with s µs = µ (s) and s s = (s) deterministic processes. Suppose moreover that, for all 0, E || + E 0 p T p # F (s) and one of assumptions (A4 ) or (A5 ) is satisfied. Then there exists a unique p pair (Y, Z) Sm [0, T ] × p m×k (0, T ) and a unique stochastic process U 2 (0, T ) such that m (a) |F (t, Yt , Zt )| dt P-a.s., 0 Yt () (b) Dom () , dP dt- a.e. (, t) × [0, T ] , (c) Ut () (Yt ()) , dP dt - a.e. (, t) × [0, T ] and for all t [0, T ] : (14) Yt + Us = + F (s, Ys , Zs ) - ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU 1 Moreover, uniqueness hol in Sm [0, T ] × 0 m×k (0, T ) , where 1 Sm [0, T ] = p>1 p Sm [0, T ] . 1+ ~ ~ Proof. Let (Y, Z), (Y , Z) Sm [0, T ] × 0 m×k (0, T ) be two solutions. p1 p2 ~ Sm [0, T ] and it follows thahen p1 , p2 > 1 such that Y Sm [0, T ], Y p ~ Y, Y Sm [0, T ], where p = p1 p2 . Applying Proposition 6 we obtain the uniqueness. To prove existence of a solution we can assume, without loss of generality, thahere exists u0 Dom () such that (15) 0 = (u0 ) (y) , y Rm , hence 0 (u0 ), since, in the sense of inition 1, we can replace BSVI (1) by ~ -dYt + (Yt ) dt F (t, Yt , Zt ) dt - Zt dBt , 0 t < T ~ YT = , where, for (u0 , u0 ) fixed, ^ (y) = (y) - (u0 ) - u0 , y - u0 , y Rd ~ ^ ~ F (t, y, z) = F (t, y, z) - u0 , y Rd , t [0, T ] . ^ Step 1. Approximating problem. Let (0, 1] and the approximating equation Yt + (16) (Ys ) = + F (s, Ys , Zs ) , t [0, T ] , is the gradient of the Yosida's regularization of the function . Using (15) we obtain (17) 0 = (u0 ) (J y) (y) (y), J (u0 ) = u0 , (u0 ) = 0. It follows from [2], Theorem 4.2 (see also [15], Chapter 5) that equation p (16) has an unique solution (Y , Z ) Sm [0, T ] × p m×k (0, T ). BSVIs WITH LOCALLY BOUNDED GENERATORS Step 2. Boundedness of Y and Z , without supplementary assumptions (A4 ) or (A5 ). From Proposition 3, applied for (16), we obtain, for all a > 1, EFt (18) + s[t,T ] T sup eVs (Ys - u0 ) e (Ys ) p/2 e |Zs |2 p/2 C EFt epVT | - u0 |p + eVs In particular there exists a constant independent of such that (a) E Y (19) T 2 T E Y (b) E |Zs |2 E p 2/p 0 C, |Zs |2 p/2 2/p Moreover, from (10) we obtain (20) |Yt | |u0 | + EFt T 1/p where is given by (13) with u0 = 0 (since (u0 ) = 0). ^ hroughouhe proof we shall fix a = 2 (and then Vt ined by (6), t with np = 1 (p - 1) = 1, becomes Vt = 0 µ (s) + 2 (s) ). Step 3. Boundedness of (Ys ). Using the following stochastic subdifferential inequality (for proof see Proposition 2.2, [13]) (Yt ) + (Ys ), dYs (YT ) = () (), we deduce that, for all t [0, T ] , (Yt ) + | (Ys )|2 () (Ys ), Zs dBs . (21) (Ys ), F (s, Ys , Zs ) - ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU Since | (Ys ) |2 |Zs |2 1/2 1 E sup |Ys | |Zs |2 1/2 1 2E sup |Ys |2 + E |Zs |2 then E t (Ys ), Zs dBs = 0. Under assumption (A4 ), since (Ys ) )), then (J (Ys (Ys ), F (s, Ys , Zs ) 1 = Y - J (Ys ) , F (s, Ys , Zs ) - F (s, J (Ys ) , Zs ) s + (Ys ), F (s, J (Ys ) , Zs ) 1 1 µ+ (s) |Ys - J (Ys )|2 + | (Ys )|2 + s + b (s) |J (Ys )|p + |Zs |2 . 2 From (2) and inequality |J (Ys )| |J (Ys ) - J (u0 )| + |u0 | |Ys - u0 | + |u0 | we have, for all t [0, T ] , 1 E (Yt ) + E 2 | (Ys )|2 E() + 2 - u0 | + |u0 |) + p |Zs |2 µ+ (s) E (Ys ) s + b (s) (|Ys that yiel, via estimate (18) and the backward Gronwall's inequality, thahere exists a constant C > 0 independent of (0, 1] such that (22) (a) E (Yt ) + E (b) | (Ys )|2 C, E |Yt - J (Yt )|2 If we suppose (A5 ) then, from (20), we infer that (23) Now (Ys ), F (s, Ys , Zs ) ), F (s, Y , 0) + (Y ), F (s, Y , Z ) - F (s, Y , 0) = (Ys s s s s s 1 )|2 + |F # (s) |2 + L2 |Z |2 . | (Ys s R0 2 |Yt ||u0 |+(EFt 2,p )1/p |u0 |+C2,p e2 T 1/p [M +|u0 |+|^0 |T ] = R0 . u BSVIs WITH LOCALLY BOUNDED GENERATORS Hence from (21) it follows that, for all t [0, T ] , 1 E(J (Yt )) + E 2 | (Ys )|2 |Zs |2 (24) E () + # |FR0 (s) |2 + L2 and from (19) we obtain boundedness inequalities (22). Step 4. Cauchy sequence and convergence. Let , (0, 1]. , We can write Yt - Yt = t dKs - t Zs dBs , where , Kt = F (s, Ys , Zs ) - F (s, Ys , Zs ) - (Ys ) + (Ys ) . Then 1 , Yt -Yt , dKt (+) (Yt ), (Yt ) dt+|Yt -Yt |2 dVt + |Zt -Zt |2 dt, 4 and by Proposition 11, with p = 2, E sup |Ys - Ys |2 + E 0 |Zs - Zs |2 CE ( + ) (Ys ), (Ys ) 1 C( + ) E 2 | (Ys )|2 + E | (Ys )|2 C ( + ). 2 2 Hence there exist (Y, Z, U ) Sm [0, T ] × 2 m×k (0, T ) × m (0, T ) and a sequence n 0 such that 2 Y n Y, in Sm [0, T ] and a.s. in C ([0, T ] ; Rm ) , 2 m×k , n Z, in 2 Z m×k (0, T ) and a.s. in L 0, T ; R (Y ) U, weakly in 2 (0, T ) , m Jn (Y n ) Y, in 2 (0, T ) and a.s. in L2 (0, T ; Rm ) . m Passing to limit in (16) we conclude that Yt + Us = + F (s, Ys , Zs ) - ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU Since (Ys ) (J (Ys )) then for all A F, 0 s t T and 2 v Sm [0, T ] , 1A (Yr ), vr - Yr dr + E t s 1A (J (Yr ))dr E 1A (vr )dr. Passing to lim inf for = n 0 in the above inequality we obtain that p 2 Us (Ys ). Hence (Y, Z, U ) Sm [0, T ] × p m×k (0, T ) × m (0, T ) and t (Y, Z, K) , with Kt = 0 Us , is the solution of BSVI (1). Step 5. Remarks in case (A5 ). Passing to lim inf for = n 0 in (23) and (24) it follows, using assumptions (A5 ), thahe solution also satisfies (a) |Yt | R0 , a.s. for all t [0, T ] , # |FR0 (s) |2 + L2 1 (b) E(Yt ) + E 2 E () + |Us |2 |Zs |2 . The proof is completed now. Remark 8. The existence Theorem 7 is well adapted to the Hilbert spaces since we do not impose an assumption of type Int (Dom ()) = , which is very restrictive for the infinite dimensional spaces. In the context of the Hilbert spaces Theorem 7 hol in the same form and one can give, as examples, partial differential backward stochastic variational inequalities (see [14]). 4. BSVI - a general existence result We replace now assumptions (A5 ) with Int (Dom ()) = . Theorem 9 (Existence). Let p 2 and assumptions (A1 -A3 ) be satisfied with s µs = µ (s) and s s = (s) deterministic processes. We suppose moreover that Int (Dom ()) = and for all 0 E ||p + E 0 T p # F (s) < . BSVIs WITH LOCALLY BOUNDED GENERATORS p Then there exists a unique triple (Y, Z, K) Sm [0, T ] × p (0, T ) × m×k p Sm (0, T ), E K T such that for all t [0, T ] : Yt + KT - Kt = + , F (s, Ys , Zs ) - (25) dKt (Yt ) dt, a.s., Y = , a.s., T p T p T p/2 p/2 which means that BSVI (1) has a unique solution, and moreover E Y K K |Zt |2 dt < . Proof. The uniqueness was proved in Proposition 6. Step 1. Existence under supplementary assumption M > 0, u0 Int(Dom()) such that (26) E|()| + || + |F (s, u0 , 0)| M, a.s. . # Let R0 ined by (23) and denote t = (t) + FR0 (t). By Theorem 7 there p exists a unique (Y n , Z n , U n ) Sm [0, T ] × p (0, T ) × 2 (0, T ) such that m m×k n (Y n ) and for all t [0, T ] : Us s (27) Ytn + n Us = + n F (s, Ysn , Zs ) 1t n - n Moreover sup |Ysn | R0 , a.s. and T p/2 (28) |(Ysn )| p/2 n |Zs |2 Let q = p/2, nq = 1 (q - 1), a = 2 and Vt2,q given by (6). Since n+l n Ytn - Ytn+l , (F (t, Ytn , Zt )1t n - Utn - F (t, Ytn+l , Zt )1t n+l + Utn+l ) dt n Ytn - Ytn+l , F (t, Ytn , Zt ) (1t n - 1t n+l )dt nq n n+l + |Ytn - Ytn+l |2 dVt2,q + |Zt - Zt |2 dt, 4 ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU then by Proposition 11, from Appendix, (with a = 2) there exists a constant depending only on p, such that E sup |Ysn Ysn+l |p/2 p/4 n |Zs n+l Zs |2 p/2 Cp ep But V 2,q n 1s n |F (s, Ysn , Zs )| p/2 n 1s n |F (s, Ysn , Zs )| # n 1s n FR0 (s) + (s) |Zs | p/2 # 1s n FR0 (s) T p/2 1/2 p/2 Cp E +Cp 0 T 1s n (s) p 1/2 # 1s n FR0 T p/2 1/2 · E |Z (s)| Cp (s) p 1/2 2 +Cp 1/2 1s n (s) p/2 0, as n . p/2 Hence there exists a pair (Y, Z) Sm [0, T ]×m×k (0, T ) such that, as n , (Y n , Z n ) (Y, Z) in Sm [0, T ]×m×k (0, T ) . In particular Y0n Y0 in · n 0 Rm and from equation (27) it follows that K·n = 0 Us K, in Sm [0, T ]. 2,p Now by (11) for Vt = Vt we obtain T p/2 p/2 p/2 |Utn | dt = E Kn p/2 Ce2p 1 + T + E ||p + E |F (t, u0 , 0)| dt with C = C (p, u0 , u0 , r0 , ) . ^ Therefore E K p/2 T Ce2p 1 + T + E ||p + E BSVIs WITH LOCALLY BOUNDED GENERATORS Passing to lim inf as n , eventually on a subsequence, we deduce from (18) and (20) that sup |Ys | R0 , a.s. and T p/2 T p/2 |(Ys )| |Zs |2 To show that (Y, Z, K) is solution of BSDE (25) it remains to show that dKt (Yt ) (dt). Applying Corollary 13 we obtain dKt (Yt ) (dt), n since dKt = Utn dt (Ytn ) dt. Step 2. Existence without supplementary assumption (26). Let (u0 , u0 ) such that u0 Int(Dom()) and B (u0 , r0 ) Dom () . ^ Recall that #0 ,r0 = sup { (u0 + r0 v) : |v| 1} < . u We introduce n = 1[0,n] (|| + | ()|) + u0 1(n,) (|| + | ()|) and F n (t, y, z) = F (s, y, z) - F (s, u0 , 0) 1|F (s,u0 ,0)|n Clearly | n | + | (n )| + |F n (t, u0 , 0)| 3n + | (u0 )| . By Step 1, for each n N there exists a p/2 p unique triple (Y n , Z n , K n ) Sm [0, T ] × p m×k (0, T ) × Sm (0, T ) solution of BSDE (29) n n Ytn + (KT - Kt ) = n + n F n (s, Ysn , Zs ) - n From Corollary 5 and Proposition 6 we infer thahere exists a constant Cp such that Er0 0 n p/2 Kn p/2 T n |Zs |2 p/2 Cp e2p V sup |(Ysn -u0 )|p |(Ysn )- (u0 )| p/2 p/2 #0 ,r0 - (u0 ) u T p/2 +|^0 |p T p u (30) | - u0 |p + E Cp e 0 2p V T |F n (s, u0 , 0)| p/2 #0 ,r0 u - (u0 ) T p/2 + |^0 |p T p + E | - u0 |p + u p/2 T . Remark that p 2 is required only to obtain the estimate of E K n ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU Since n n+l Ysn - Ysn+l , F n (s, Ysn , Zs ) - F n+l (s, Ysn+l , Zs ) |Ysn - Ysn+l ||F (s, u0 , 0) |1|F (s,u0 ,0)|n 1 n n+l + |Ysn - Ysn+l |2 dVt + |Zs - Zs |2 4 then by Proposition 11 we obtain E sup |Ysn - Ysn+l |p + E n n+l |Zs - Zs |2 p/2 Cp e2p E | - u0 |p 1||+|()|n |F (s, u0 , 0)| 1|F (s,u0 ,0)|n p n n Hence there exists a pair (Y, Z) Sm [0, T ]×p m×k (0, T ) such that (Y , Z ) p p (Y, Z), as n , in Sm [0, T ] × m×k (0, T ). In particular Y0n Y0 in 0 Rm . From equation (29) we have K n K in Sm [0, T ] , and for all t [0, T ] Yt + KT - Kt = + F (s, Ys , Zs ) - Letting n and applying Proposition 12 we can asserhat estimate (30) n hol without n. To complete the proof remark that from dKt (Ytn ) dt we can infer, using Corollary 13, that dKt (Yt ) dt. Therefore (Y, Z, K) is solution of BSDE (25) in the sense of inition 1. Remark 10. When µ and are stochastic processes we obtain, with similar proofs as in Theorems 7 and 9, the existence of a solution in the space Up (0, T ) = m,k where (Y, Z) p = 0 (Y, Z)Sm [0, T ]×0 (0, T ) : (Y, Z) m×k <, a > 1 , sup epVs |Ys |p + E 0 e |Zs |2 p/2 BSVIs WITH LOCALLY BOUNDED GENERATORS 5. Appendix In this section we first present some useful and general estimates on 0 (Y, Z) Sm [0, T ] × 0 m×k (0, T ) satisfying an identity of type Yt = YT + dKs - Zs dBs , t [0, T ] , P-a.s., 0 where K Sm [0, T ] and K· () BV ([0, T ] ; Rm ) P-a.s., . The following results and their proofs are given in the monograph of s Pardoux, Ra¸canu [15], Annex Assume there exist D, R, N progressively measurable increasing continuous stochastic processes with D0 = R0 = N0 = 0, V progressively measurable bounded-variation continuous stochastic process with V0 = 0, a, p > 1, such that, as signed measures on [0, T ] , (31) dDt + Yt , dKt 1p2 dRt + |Yt |dNt + |Yt |2 dVt + np |Zt |2 dt, 2a . where np = (p - 1) 1. Let Y eV [t,T ] = sup Ys eVs and Y eV s[t,T ] T Y eV [0,T ] Proposition 11. Assume (31) and E Ye V p p/2 1p2 dRs e dNs Vs < . Then there exists a positive constant C , depending only of a, p, such that, P-a.s., for all t [0, T ] : EFt (32) sup |eVs Ys |p + s[t,T ] T e d p/2 e |Zs |2 p/2 + EFt epVs |Ys |p-2 1Ys =0 d + eVT YT p T epVs |Ys |p-2 1Ys =0 |Zs |2 p/2 T C EFt e 1p2 dRs eVs dNs ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU In particular for all t [0, T ] : |Yt |p C EFt p p |YT |p + 1p2 RT + NT e p (V· -Vt )+ [t,T ] , P-a.s. Moreover if there exists a constant b 0 such that for all t [0, T ] : e VT -V 1/2 YT + 2(Vs -Vt ) 1p2 dRs e(Vs -Vt ) dNs b, a.s. then for all t [0, T ] : (33) |Yt |p + EFt p/2 e2(Vs -Vt ) |Zs |2 bp C , P-a.s. The following results provides a criterion for passing to the limit in Stieltjes integral (for the proofs we refer the reader to [15], Chapter I). Proposition 12. Let Y, K, Y n , K n be C ([0, T ] ; Rm )-valued random variables, n N. Assume (i) p > 0 such that sup E K n nN Kn - K p T (ii) ( Y n - Y T + - T ) - - 0, as n , n-Y n i.e. > 0, P {( Y T + K - K T ) > } 0, as n . Then, for all 0 s t T : t s n Yrn , dKr - - - p T prob. t prob. Yr , dKr , as n , s p T and moreover, E K lim inf n+ E K n Corollary 13. Lehe assumptions of Proposition 12 be satisfied. If A : Rm Rm is a (multivalued) maximal monotone operator then the following implication hol n dKt A (Ytn ) dt on [0, T ] , a.s. dKt A (Yt ) dt on [0, T ] , a.s. In particular if : Rd ] - , +] is a proper convex l.s. function then n dKt (Ytn ) dt on [0, T ] , a.s. dKt (Yt ) dt on [0, T ] , a.s. BSVIs WITH LOCALLY BOUNDED GENERATORS http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of the Alexandru Ioan Cuza University - Mathematics de Gruyter

Backward Stochastic Variational Inequalities with Locally Bounded Generators

Download PDF
24 pages

Loading next page...
 
/lp/de-gruyter/backward-stochastic-variational-inequalities-with-locally-bounded-TjoFyk0pUP

References (15)

Publisher
de Gruyter
Copyright
Copyright © 2014 by the
ISSN
1221-8421
eISSN
1221-8421
DOI
10.2478/aicu-2013-0023
Publisher site
See Article on Publisher Site

Abstract

The paper deals with the existence and uniqueness of the solution of the backward stochastic variational inequality: -dYt + (Yt ) dt F (t, Yt , Zt ) dt - Zt dBt , 0 t < T YT = , where F satisfies a local boundedness condition. Mathematics Subject Classification 2010: 60H10, 93E03, 47J20, 49J40. Key wor: backward stochastic differential equations, subdifferential operators, stochastic variational inequalities. 1. Introduction We consider the following backward stochastic variational inequality (BSVI): (1) -dYt + (Yt ) dt F (t, Yt , Zt ) dt - Zt dBt , 0 t < T YT = , where {Bt : t 0} is a standard Brownian motion, is the subdifferential of a convex l.s. function , and T > 0 is a fixed deterministic time. The study of the backward stochastic differential equations (BSDE) (equation of type (1) withouhe subdifferential operator) was initiated * The work was supported by IDEAS project, no. DRU/89/1.5/S/49944 project. 241/05.10.2011 and by POS- ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU by Pardoux and Peng in [11] (see also [12]) where is proved the existence and the uniqueness of the solution for the BSDE under the assumption of Lipschitz continuity of F with respeco y and z and square integrability of and F (t, 0, 0). The more general case of scalar BSDE with one-sided reflection and associated optimal control problems was considered by El Karoui et al. [8] and with two-sided reflection associated with stochastic game problem by Cvitanic, Karatzas [6] (see also [3] and [7] for the investigation of zero-sum two-player stochastic differential games whose cost functionals are given by controlled reflected BSDE). On the other hand, it is worth to mention the backward in time problems in mechanics of continua, since a large number of physical phenomena lea to these new non-standard problems. Specify that for improperly posed problems the solutions will not exist for arbitrary data and not depend continuously on the data (see e.g. [5], [4] and references therein). s The standard work on BSVI is that of Pardoux and Ra¸canu [13], which give a proof of existence and uniqueness of the solution for (1) under the following assumptions on F : monotonicity with respeco y (in the sense that y - y, F (t, y , z) - F (t, y, z) |y - y|2 ), Lipschitzianity with respeco z and a sublinear growth for F (t, y, 0), |F (t, y, 0)| t + L |y| , (t, y) [0, T ] × Rm. It is proved thahere exists a unique triple (Y, Z, K) such that Yt +KT -Kt =+ F (s, Ys , Zs ) - , with dKt (Yt ) dt. Moreover the process K is absolute continuous with respeco dt. In [14] the same authors extend the results from [13] to the Hilbert spaces framework. Using a mixed Euler-Yosida scheme, Maticiuc, Rotenstein provided in [9] numerical results concerning the multi-valued stochastic differential equation (1). Our paper generalize the previous existence and uniqueness results for (1) by assuming a local boundedness condition (instead of sublinear growth of F ), i.e. # F (s) # where F (t) = sup |F (t, y, 0)| . |y| Concerning to this requirement on F we remark that a similar one was considered by Pardoux in [10] for the study of BSDE. More precisely, his BSVIs WITH LOCALLY BOUNDED GENERATORS result is the following: if L2 (; Rm ), F (t, 0, 0) L2 ( × [0, T ] ; Rm ), F is monotone with respeco y, Lipschitz with respeco z and there exists a deterministic continuous increasing function such that (t, y) [0, T ] × Rm , |F (t, y, 0)| |F (t, 0, 0)| + (|y|) , P-a.s, then there exist a unique solution for BSDE (1) with 0. This result was generalized by Briand et al. [2]. The article is organized as follows: in the next Section we prove some a priori estimates and the uniqueness result for the solution of BSVI (1). Section 3 is concerned on the existence result under two alternative assumptions (which allow to obtain the absolute continuity of the process K) and Section 4 establishes the general existence result. In the Appendix we presents, following [15], some results useful throughouhe paper. 2. Preliminaries; a priori estimates and the uniqueness result Let {Bt : t 0} be a k-dimensional standard Brownian motion ined on some complete probability space (, F, P). We denote by {Ft : t 0} the natural filtration generated by {Bt : t 0} and augmented by N , the set of P- null events of F, Ft = {Br : 0 r t} N . We suppose thahe following assumptions hol: (A1 ) : Rm is a FT -measurable random vector, (A2 ) F : × [0, T ] × Rm × Rm×k Rm satisfies that, for all y Rm , z Rm×k , (, t) - F (·, ·, y, z) : × [0, T ] Rm is progressively measurable stochastic process, and there exist µ : × [0, T ] R and : × [0, T ] R+ progressively measurable stochastic processes T with 0 |µt | + 2 dt such that, for all t [0, T ], y, y Rm t and z, z Rm×k , P-a.s.: (Cy ) y - F (t, y, z) : Rm Rm is continuous, (My ) y - y, F (t, y , z) - F (t, y, z) µt |y - y|2 , (Lz ) |F (t, y, z ) - F (t, y, z)| t |z - z|, (By ) # F (s) 0, # where, for 0, F (t) = sup |F (t, y, 0)| , |y| (A3 ) : Rm (-, +] is a proper, convex l.s. function. ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU The subdifferential of is given by (y) = {^ Rm : y , v-y +(y) y ^ (v), v Rm }. We ine Dom() = {y Rm : (y) < }, Dom() = {y Rm : (y) = } Dom() and by (y, y ) we understand that y Dom () ^ and y (y). ^ Recall that Dom() = Dom(), Int(Dom()) = Int(Dom()). Let > 0 and the Moreau-Yosida regularization of : (y) = inf (2) = 1 |y - v|2 + (v) : v Rm 2 1 |y - J (y)|2 + (J (y)) , 2 where J (y) = (Im×m + )-1 (y). Remark that is a C 1 convex function and J is a 1-Lipschitz function. s We mention some properties (see Br´zis [1], and Pardoux, Ra¸canu e [13] for the last one): for all x, y Rm (a) (y) = (y) = (3) y - J (y) (J y), 1 (b) | (x) - (y)| |x - y| , (c) (x) - (y), x - y 0, (d) (x) - (y), x - y -( + ) (x), (y) . p We denote by Sm [0, T ] the space of (equivalent classes of) progressively measurable and continuous stochastic processes X : × [0, T ] Rm such that E supt[0,T ] |Xt |p if p > 0, and by p (0, T ) the space of (equivalent m classes of) progressively measurable stochastic process X : × [0, T ] Rm such tha |Xt |2 dt P-a.s. , if p = 0, T p/2 |Xt | dt if p > 0. For a function g : [0, T ] Rm , let us denote by g g on [0, T ] i.e. g T n-1 the total variation of = sup i=0 |g (ti+1 ) - g (ti )| : n N , 0 = t0 < t1 < · · · tn = T BSVIs WITH LOCALLY BOUNDED GENERATORS and by BV ([0, T ] ; Rm ) the space of the functions g : [0, T ] Rm such that g T < (BV ([0, T ] ; Rm ) equipped with the norm ||g||BV ([0,T ];Rm ) = |g(0)| + g T is a Banach space). 0 inition 1. A pair (Y, Z) Sm [0, T ] × 0 m×k (0, T ) of stochastic processes is a solution of backward stochastic variational inequality (1) if 0 there exists K Sm [0, T ] with K0 = 0, such tha (a) (b) 0 s | (Yt )| dt + |F (t, Yt , Zt )| dt a.s., s s dKt (Yt ) dt, a.s. that is: P-a.s., y(r) - Yr , dKr + (Yr )dr (y(r))dr, y C([0, T ]; Rd ), 0 t s T, and, P-a.s., for all t [0, T ] : (4) Yt + KT - Kt = + F (s, Ys , Zs ) - Zs dBs (we also say thariplet (Y, Z, K) is solution of equation (1)). Remark 2. If K is absolute continuous with respeco dt, i.e. there exists a progressively measurable stochastic process U such that |Ut | dt a.s. and Kt = Us , for all t [0, T ] , then dKt (Yt ) dt means Ut (Yt ) , dt-a.e., a.s. ~ ~ If dKt (Yt ) dt and dKt (Yt )dhen we clearly have | (Yt )| dt + ~ |(Yt )|dt a.s. and, using the subdifferential inequalities s t ~ Yr - Yr , dKr + (Yr )dr s t ~ (Yr )dr, (Yr )dr, ~ ~ Yr - Yr , dKr + ~ (Yr )dr ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU we infer that, for all 0 t s T (5) ~ ~ Yr - Yr , dKr - dKr 0, a.s. Let a, p > 1 and (6) Vt = Vt = 0 t µs + a 2 , 2np s where np = (p - 1) 1. Denote 1 Sm + ,p [0, T ] = 0 Y Sm [0, T ] : a > 1, E sup |eVs Ys |p < . Remark that if µs and 2 are deterministic functions then, for all p > 1, s 1+ p Sm ,p [0, T ] = Sm [0, T ]. Proposition 3. Let (u0 , u0 ) and assumptions (A1 -A3 ) be satis^ fied. Then for every a, p > 1 there exists a constant C such that for every (Y, Z) solution of BSDE (1) satisfying E sup epVs |Ys - u0 |p + E 0 T u eVs (|^0 | + |F (s, u0 , 0)|) the following inequality hol P-a.s., for all t [0, T ] : E Ft sup e s[t,T ] Vs (Ys - u0 ) + p/2 |Zs | p/2 Ft (7) Ft F e |(Ys ) - (u0 )| epVs |Ys - u0 |p-2 1Ys =u0 |Zs |2 epVs |Ys - u0 |p-2 1Ys =u0 |(Ys ) - (u0 )| C EFt epVT | - u0 |p T p u eVs |^0 | eVs BSVIs WITH LOCALLY BOUNDED GENERATORS and, for every R0 > 0 and p 2, p/2 R0 EF p/2 |F (s, Ys , Zs )| Ft (8) epVs |Ys - u0 |p-2 1Ys =u0 |F (s, Ys , Zs )| p/2 p/2 C EFt epVT | - u0 |p + R0 ·E F 1p2 # Fu0 ,R0 (s) + + R0 s Ft where # eVs Fu0 ,R0 (s) + 2R0 |s | # Fu0 ,R0 (t) = sup |y-u0 |R0 |F (t, y, 0)| . Proof. We can write Yt - u0 = - u0 + [F (s, Ys , Zs ) - dKs ] - Zs dBs . Let R0 0. The monotonicity property of F implies that, for all |v| 1 : F (t, u0 + R0 v, z) - F (t, y, z) , u0 + R0 v - y µt |u0 + R0 v - y|2 , and, consequently R0 F (t, y, z) , -v + F (t, y, z) , y - u0 µt |u0 + R0 v - y|2 + |F (t, u0 + R0 v, z)| |y - R0 v - u0 | # µt |u0 + R0 v - y|2 + Fu0 ,R0 (t) + t |z| |y - R0 v - u0 | # µt |u0 + R0 v - y|2 + Fu0 ,R0 (t) |y - R0 v - u0 | np 2 a 2 # + |z| Fu0 ,R0 (t) (|y - u0 | + R0 ) t |y - R0 v - u0 |2 + 2np 2a np # 2 2 + + t |y - u0 |2 -2R0 v, y - u0 +R0 |v|2 + |z|2 R0 Fu0 ,R0 (t) + R0 t 2a np 2 # |z| . + Fu0 ,R0 (t) + 2R0 |t | |y - u0 | + t |y - u0 |2 + 2a ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU Taking sup|v|1 , we have R0 |F (t, Yt , Zt )| dt + Yt - u0 , F (t, Yt , Zr ) dt # # 2 + R0 Fu0 ,R0 (t) + R0 t + Fu0 ,R0 (t) + 2R0 |t | |Yt - u0 | np |Zt |2 . + |Yt - u0 |2 dVt + 2a From the subdifferential inequalities we have |(t, Yt )-(t, u0 )| [(t, Yt )- (t, u0 )]+2|^0 ||Yt -u0 | and [(t, Yt )-(t, u0 )]dt Yt -u0 , dKt . Therefore u |(t, Yt ) - (t, u0 )|dt Yt - u0 , dKt + 2|^0 ||Yt - u0 |dt. From the above it u follows that (9) [R0 |F (t, Yt , Zt )| + |(Yt )- (u0 )|] dt+ Yt -u0 , F (t, Yt , Zt ) dt-dKt # # 2 + R0 Fu0 ,R0 (t) +R0 t dt+ Fu0 ,R0 (t) +2R0 |t | +2 |^0 | |Yt -u0 | dt u np 2 2 |Zt | . + |Yt - u0 | dVt + 2a For R0 = 0, inequality (7) clearly follows from (9) applying Proposition 11 from Appendix. For R0 > 0 we moreover deduce, using once again Proposition 11, inequality (8). Remark 4. Denoting = eVT | - u0 | + 0 0 eVs |^0 | + u eVs we deduce that, for all t [0, T ] : (10) 1/p |Yt | |u0 | + C e-Vt EFt p 1/p , a.s. Corollary 5. Let p 2. We suppose moreover thahere exist r0 , c0 > 0 such that #0 ,r0 = sup { (u0 + r0 v) : |v| 1} c0 . u Then r0 EFt (11) C E + t Ft p/2 p/2 e d K s p pVT | - u0 | + #0 ,r0 u p/2 - (u0 ) t p eVs |^0 | u eVs BSVIs WITH LOCALLY BOUNDED GENERATORS Proof. Let an arbitrary function v C ([0, T ] ; Rm ) such that v T 1. From the subdifferential inequality u0 + r0 v (t) - Yt , dKt + (Yt )dt (u0 + r0 v (t)) dt, we deduce that r0 d K t + (Yt )dt Yt - u0 , dKt + #0 ,r0 dt. Since Yt - u0 , u0 + (u0 ) (Yt ), then ^ u r0 d K Therefore r0 d K t + Yt - u0 , F (t, Yt , Zt ) dt - dKt #0 ,r0 - (u0 ) dt + |Yt - u0 | (|^0 | + |F (t, u0 , 0)|) dt u u np 2 2 |Zt | dt. + |Yt - u0 | dVt + 2a The inequality (11) follows using Proposition 11. Proposition 6 (Uniqueness). Let assumptions (A1 -A3 ) be satisfied. 0 ~ ~ Let a, p > 1. If (Y, Z) , (Y , Z) Sm [0, T ] × 0 m×k (0, T ) are two solutions of BSDE (1) corresponding respectively to and such that E sup epVs |Ys - ~ ~ ~s |p then for all t [0, T ] , epVt |Ys - Ys |p EFt epVT | - |p , P-a.s. ~ Y and there exists a constant C such that P-a.s., for all t [0, T ] : EFt ~ sup epVs |Ys - Ys |p + s[t,T ] Yt - u0 , dKt + |^0 | |Yt - u0 | dt + #0 ,r0 - (u0 ) dt. u u (12) ~ e |Zs - Zs |2 p/2 ~ C EFt epVT | - |p . + ,p 1 Moreover, the uniqueness of solution (Y, Z) of BSDE (1) hol in Sm 0 m×k (0, T ). [0, T ]× 0 ~ ~ Proof. Let (Y, Z), (Y , Z) Sm [0, T ] × 0 m×k (0, T ) be two solutions corresponding to and respectively. Then there exists p > 1 such that ~ p ~ ~ ~ Y, Y Sm [0, T ] and Yt - Yt = - + t dLs - t (Zs - Zs )dBs where ~ t ~ ~ ~ Lt = 0 [(F (s, Ys , Zs ) - F (s, Ys , Zs )) - (dKs - dKs )]. Since by (5) Ys - ~ ~ Ys , dKs - dKs 0, then, for all a > 1, ~ ~ ~ ~ Yt - Yt , dLt |Yt - Yt |2 µt dt + |Yt - Yt ||Zt - Zt |t dt np a 2 ~ ~ dt + |Zt - Zt |2 dt. |Yt - Yt |2 µt + 2np t 2a By Proposition 11, from Appendix, inequality (12) follows. ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU 1+ ~ ~ Let now p > 1 be such that (Y, Z) , (Y , Z) Sm ,p [0, T ] × 0 m×k (0, T ) are two solutions of BSDE (1) corresponding respectively to and . From ~ 1+,p the inition of space Sm [0, T ] there exists a > 1 such that ~ E sup |eVt Yt |p E sup |eVt Yt |p < . t[0,T ] t[0,T ] Consequently estimate (12) follows and uniqueness too. 3. BSVI - an existence result Using Proposition 3 we can prove now the existence of a triple (Y, Z, K) which is a solution, in the sense of inition 1, for BSVI (1). In order to obtain the absolute continuity with respeco dt for the process K it is necessary to impose a supplementary assumption. Let (u0 , u0 ) be fixed and ^ = C e2p T T V | - u0 |p + 0 p |^0 | u (13) where a, p > 1, C is the constant given by Proposition 3 and Vt is ined by (6). T If there exists a constant M such that || + 0 M, a.s. p p p u then T C e2p V T [(M + |u0 |) + |^0 | T ] and by (10) |Yt | |u0 | + EFt T 1/p 1/p |u0 | + C e2 V [M + |u0 | + |^0 | T ] , a.s. u We will make the following assumptions: (A4 ) There exist p 2, a positive stochastic process L1 ( × (0, T )), a positive function b L1 (0, T ) and a real number 0, such that (i) E+ () (ii) for all (u, u) and z Rm×k : ^ 1 u u, F (t, u, z) |^|2 + t + b (t) |u|p + |z|2 ^ 2 dP dt-a.e., (, t) × [0, T ] , and BSVIs WITH LOCALLY BOUNDED GENERATORS (A5 ) There exist M, L > 0 and (u0 , u0 ) such that: ^ (i) (ii) E+ () t L, a.e., t [0, T ] , (iii) || + M, a.s., , 1/p V (iv) R0 |u0 | + C e2 [M + |u0 | + |^0 | T ] u such that E # FR0 (s) < . We note that, if u, F (t, u, z) 0, for all (u, u) , then condition ^ ^ (A4 -ii) is satisfied with t = b (t) = = 0. For example, if = ID (the ¯ ¯ convex indicator of closed convex set D) and ny denotes the unit outward ¯ ¯ normal vector to D at y Bd D , then condition ny , F (t, y, z) 0 for ¯ yiel (A4 -ii) with t = b (t) = = 0. In this last case the all y Bd D It^'s formula for (y) = [distD (y)]2 and the uniqueness yiel K = 0. o ¯ We also remark that if F (t, y, z) = F (y, z) then assumptions (A5 ) becomes || + E+ () M, a.s., . Theorem 7 (Existence). Let p 2 and assumptions (A1 -A3 ) be satisfied with s µs = µ (s) and s s = (s) deterministic processes. Suppose moreover that, for all 0, E || + E 0 p T p # F (s) and one of assumptions (A4 ) or (A5 ) is satisfied. Then there exists a unique p pair (Y, Z) Sm [0, T ] × p m×k (0, T ) and a unique stochastic process U 2 (0, T ) such that m (a) |F (t, Yt , Zt )| dt P-a.s., 0 Yt () (b) Dom () , dP dt- a.e. (, t) × [0, T ] , (c) Ut () (Yt ()) , dP dt - a.e. (, t) × [0, T ] and for all t [0, T ] : (14) Yt + Us = + F (s, Ys , Zs ) - ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU 1 Moreover, uniqueness hol in Sm [0, T ] × 0 m×k (0, T ) , where 1 Sm [0, T ] = p>1 p Sm [0, T ] . 1+ ~ ~ Proof. Let (Y, Z), (Y , Z) Sm [0, T ] × 0 m×k (0, T ) be two solutions. p1 p2 ~ Sm [0, T ] and it follows thahen p1 , p2 > 1 such that Y Sm [0, T ], Y p ~ Y, Y Sm [0, T ], where p = p1 p2 . Applying Proposition 6 we obtain the uniqueness. To prove existence of a solution we can assume, without loss of generality, thahere exists u0 Dom () such that (15) 0 = (u0 ) (y) , y Rm , hence 0 (u0 ), since, in the sense of inition 1, we can replace BSVI (1) by ~ -dYt + (Yt ) dt F (t, Yt , Zt ) dt - Zt dBt , 0 t < T ~ YT = , where, for (u0 , u0 ) fixed, ^ (y) = (y) - (u0 ) - u0 , y - u0 , y Rd ~ ^ ~ F (t, y, z) = F (t, y, z) - u0 , y Rd , t [0, T ] . ^ Step 1. Approximating problem. Let (0, 1] and the approximating equation Yt + (16) (Ys ) = + F (s, Ys , Zs ) , t [0, T ] , is the gradient of the Yosida's regularization of the function . Using (15) we obtain (17) 0 = (u0 ) (J y) (y) (y), J (u0 ) = u0 , (u0 ) = 0. It follows from [2], Theorem 4.2 (see also [15], Chapter 5) that equation p (16) has an unique solution (Y , Z ) Sm [0, T ] × p m×k (0, T ). BSVIs WITH LOCALLY BOUNDED GENERATORS Step 2. Boundedness of Y and Z , without supplementary assumptions (A4 ) or (A5 ). From Proposition 3, applied for (16), we obtain, for all a > 1, EFt (18) + s[t,T ] T sup eVs (Ys - u0 ) e (Ys ) p/2 e |Zs |2 p/2 C EFt epVT | - u0 |p + eVs In particular there exists a constant independent of such that (a) E Y (19) T 2 T E Y (b) E |Zs |2 E p 2/p 0 C, |Zs |2 p/2 2/p Moreover, from (10) we obtain (20) |Yt | |u0 | + EFt T 1/p where is given by (13) with u0 = 0 (since (u0 ) = 0). ^ hroughouhe proof we shall fix a = 2 (and then Vt ined by (6), t with np = 1 (p - 1) = 1, becomes Vt = 0 µ (s) + 2 (s) ). Step 3. Boundedness of (Ys ). Using the following stochastic subdifferential inequality (for proof see Proposition 2.2, [13]) (Yt ) + (Ys ), dYs (YT ) = () (), we deduce that, for all t [0, T ] , (Yt ) + | (Ys )|2 () (Ys ), Zs dBs . (21) (Ys ), F (s, Ys , Zs ) - ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU Since | (Ys ) |2 |Zs |2 1/2 1 E sup |Ys | |Zs |2 1/2 1 2E sup |Ys |2 + E |Zs |2 then E t (Ys ), Zs dBs = 0. Under assumption (A4 ), since (Ys ) )), then (J (Ys (Ys ), F (s, Ys , Zs ) 1 = Y - J (Ys ) , F (s, Ys , Zs ) - F (s, J (Ys ) , Zs ) s + (Ys ), F (s, J (Ys ) , Zs ) 1 1 µ+ (s) |Ys - J (Ys )|2 + | (Ys )|2 + s + b (s) |J (Ys )|p + |Zs |2 . 2 From (2) and inequality |J (Ys )| |J (Ys ) - J (u0 )| + |u0 | |Ys - u0 | + |u0 | we have, for all t [0, T ] , 1 E (Yt ) + E 2 | (Ys )|2 E() + 2 - u0 | + |u0 |) + p |Zs |2 µ+ (s) E (Ys ) s + b (s) (|Ys that yiel, via estimate (18) and the backward Gronwall's inequality, thahere exists a constant C > 0 independent of (0, 1] such that (22) (a) E (Yt ) + E (b) | (Ys )|2 C, E |Yt - J (Yt )|2 If we suppose (A5 ) then, from (20), we infer that (23) Now (Ys ), F (s, Ys , Zs ) ), F (s, Y , 0) + (Y ), F (s, Y , Z ) - F (s, Y , 0) = (Ys s s s s s 1 )|2 + |F # (s) |2 + L2 |Z |2 . | (Ys s R0 2 |Yt ||u0 |+(EFt 2,p )1/p |u0 |+C2,p e2 T 1/p [M +|u0 |+|^0 |T ] = R0 . u BSVIs WITH LOCALLY BOUNDED GENERATORS Hence from (21) it follows that, for all t [0, T ] , 1 E(J (Yt )) + E 2 | (Ys )|2 |Zs |2 (24) E () + # |FR0 (s) |2 + L2 and from (19) we obtain boundedness inequalities (22). Step 4. Cauchy sequence and convergence. Let , (0, 1]. , We can write Yt - Yt = t dKs - t Zs dBs , where , Kt = F (s, Ys , Zs ) - F (s, Ys , Zs ) - (Ys ) + (Ys ) . Then 1 , Yt -Yt , dKt (+) (Yt ), (Yt ) dt+|Yt -Yt |2 dVt + |Zt -Zt |2 dt, 4 and by Proposition 11, with p = 2, E sup |Ys - Ys |2 + E 0 |Zs - Zs |2 CE ( + ) (Ys ), (Ys ) 1 C( + ) E 2 | (Ys )|2 + E | (Ys )|2 C ( + ). 2 2 Hence there exist (Y, Z, U ) Sm [0, T ] × 2 m×k (0, T ) × m (0, T ) and a sequence n 0 such that 2 Y n Y, in Sm [0, T ] and a.s. in C ([0, T ] ; Rm ) , 2 m×k , n Z, in 2 Z m×k (0, T ) and a.s. in L 0, T ; R (Y ) U, weakly in 2 (0, T ) , m Jn (Y n ) Y, in 2 (0, T ) and a.s. in L2 (0, T ; Rm ) . m Passing to limit in (16) we conclude that Yt + Us = + F (s, Ys , Zs ) - ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU Since (Ys ) (J (Ys )) then for all A F, 0 s t T and 2 v Sm [0, T ] , 1A (Yr ), vr - Yr dr + E t s 1A (J (Yr ))dr E 1A (vr )dr. Passing to lim inf for = n 0 in the above inequality we obtain that p 2 Us (Ys ). Hence (Y, Z, U ) Sm [0, T ] × p m×k (0, T ) × m (0, T ) and t (Y, Z, K) , with Kt = 0 Us , is the solution of BSVI (1). Step 5. Remarks in case (A5 ). Passing to lim inf for = n 0 in (23) and (24) it follows, using assumptions (A5 ), thahe solution also satisfies (a) |Yt | R0 , a.s. for all t [0, T ] , # |FR0 (s) |2 + L2 1 (b) E(Yt ) + E 2 E () + |Us |2 |Zs |2 . The proof is completed now. Remark 8. The existence Theorem 7 is well adapted to the Hilbert spaces since we do not impose an assumption of type Int (Dom ()) = , which is very restrictive for the infinite dimensional spaces. In the context of the Hilbert spaces Theorem 7 hol in the same form and one can give, as examples, partial differential backward stochastic variational inequalities (see [14]). 4. BSVI - a general existence result We replace now assumptions (A5 ) with Int (Dom ()) = . Theorem 9 (Existence). Let p 2 and assumptions (A1 -A3 ) be satisfied with s µs = µ (s) and s s = (s) deterministic processes. We suppose moreover that Int (Dom ()) = and for all 0 E ||p + E 0 T p # F (s) < . BSVIs WITH LOCALLY BOUNDED GENERATORS p Then there exists a unique triple (Y, Z, K) Sm [0, T ] × p (0, T ) × m×k p Sm (0, T ), E K T such that for all t [0, T ] : Yt + KT - Kt = + , F (s, Ys , Zs ) - (25) dKt (Yt ) dt, a.s., Y = , a.s., T p T p T p/2 p/2 which means that BSVI (1) has a unique solution, and moreover E Y K K |Zt |2 dt < . Proof. The uniqueness was proved in Proposition 6. Step 1. Existence under supplementary assumption M > 0, u0 Int(Dom()) such that (26) E|()| + || + |F (s, u0 , 0)| M, a.s. . # Let R0 ined by (23) and denote t = (t) + FR0 (t). By Theorem 7 there p exists a unique (Y n , Z n , U n ) Sm [0, T ] × p (0, T ) × 2 (0, T ) such that m m×k n (Y n ) and for all t [0, T ] : Us s (27) Ytn + n Us = + n F (s, Ysn , Zs ) 1t n - n Moreover sup |Ysn | R0 , a.s. and T p/2 (28) |(Ysn )| p/2 n |Zs |2 Let q = p/2, nq = 1 (q - 1), a = 2 and Vt2,q given by (6). Since n+l n Ytn - Ytn+l , (F (t, Ytn , Zt )1t n - Utn - F (t, Ytn+l , Zt )1t n+l + Utn+l ) dt n Ytn - Ytn+l , F (t, Ytn , Zt ) (1t n - 1t n+l )dt nq n n+l + |Ytn - Ytn+l |2 dVt2,q + |Zt - Zt |2 dt, 4 ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU then by Proposition 11, from Appendix, (with a = 2) there exists a constant depending only on p, such that E sup |Ysn Ysn+l |p/2 p/4 n |Zs n+l Zs |2 p/2 Cp ep But V 2,q n 1s n |F (s, Ysn , Zs )| p/2 n 1s n |F (s, Ysn , Zs )| # n 1s n FR0 (s) + (s) |Zs | p/2 # 1s n FR0 (s) T p/2 1/2 p/2 Cp E +Cp 0 T 1s n (s) p 1/2 # 1s n FR0 T p/2 1/2 · E |Z (s)| Cp (s) p 1/2 2 +Cp 1/2 1s n (s) p/2 0, as n . p/2 Hence there exists a pair (Y, Z) Sm [0, T ]×m×k (0, T ) such that, as n , (Y n , Z n ) (Y, Z) in Sm [0, T ]×m×k (0, T ) . In particular Y0n Y0 in · n 0 Rm and from equation (27) it follows that K·n = 0 Us K, in Sm [0, T ]. 2,p Now by (11) for Vt = Vt we obtain T p/2 p/2 p/2 |Utn | dt = E Kn p/2 Ce2p 1 + T + E ||p + E |F (t, u0 , 0)| dt with C = C (p, u0 , u0 , r0 , ) . ^ Therefore E K p/2 T Ce2p 1 + T + E ||p + E BSVIs WITH LOCALLY BOUNDED GENERATORS Passing to lim inf as n , eventually on a subsequence, we deduce from (18) and (20) that sup |Ys | R0 , a.s. and T p/2 T p/2 |(Ys )| |Zs |2 To show that (Y, Z, K) is solution of BSDE (25) it remains to show that dKt (Yt ) (dt). Applying Corollary 13 we obtain dKt (Yt ) (dt), n since dKt = Utn dt (Ytn ) dt. Step 2. Existence without supplementary assumption (26). Let (u0 , u0 ) such that u0 Int(Dom()) and B (u0 , r0 ) Dom () . ^ Recall that #0 ,r0 = sup { (u0 + r0 v) : |v| 1} < . u We introduce n = 1[0,n] (|| + | ()|) + u0 1(n,) (|| + | ()|) and F n (t, y, z) = F (s, y, z) - F (s, u0 , 0) 1|F (s,u0 ,0)|n Clearly | n | + | (n )| + |F n (t, u0 , 0)| 3n + | (u0 )| . By Step 1, for each n N there exists a p/2 p unique triple (Y n , Z n , K n ) Sm [0, T ] × p m×k (0, T ) × Sm (0, T ) solution of BSDE (29) n n Ytn + (KT - Kt ) = n + n F n (s, Ysn , Zs ) - n From Corollary 5 and Proposition 6 we infer thahere exists a constant Cp such that Er0 0 n p/2 Kn p/2 T n |Zs |2 p/2 Cp e2p V sup |(Ysn -u0 )|p |(Ysn )- (u0 )| p/2 p/2 #0 ,r0 - (u0 ) u T p/2 +|^0 |p T p u (30) | - u0 |p + E Cp e 0 2p V T |F n (s, u0 , 0)| p/2 #0 ,r0 u - (u0 ) T p/2 + |^0 |p T p + E | - u0 |p + u p/2 T . Remark that p 2 is required only to obtain the estimate of E K n ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU Since n n+l Ysn - Ysn+l , F n (s, Ysn , Zs ) - F n+l (s, Ysn+l , Zs ) |Ysn - Ysn+l ||F (s, u0 , 0) |1|F (s,u0 ,0)|n 1 n n+l + |Ysn - Ysn+l |2 dVt + |Zs - Zs |2 4 then by Proposition 11 we obtain E sup |Ysn - Ysn+l |p + E n n+l |Zs - Zs |2 p/2 Cp e2p E | - u0 |p 1||+|()|n |F (s, u0 , 0)| 1|F (s,u0 ,0)|n p n n Hence there exists a pair (Y, Z) Sm [0, T ]×p m×k (0, T ) such that (Y , Z ) p p (Y, Z), as n , in Sm [0, T ] × m×k (0, T ). In particular Y0n Y0 in 0 Rm . From equation (29) we have K n K in Sm [0, T ] , and for all t [0, T ] Yt + KT - Kt = + F (s, Ys , Zs ) - Letting n and applying Proposition 12 we can asserhat estimate (30) n hol without n. To complete the proof remark that from dKt (Ytn ) dt we can infer, using Corollary 13, that dKt (Yt ) dt. Therefore (Y, Z, K) is solution of BSDE (25) in the sense of inition 1. Remark 10. When µ and are stochastic processes we obtain, with similar proofs as in Theorems 7 and 9, the existence of a solution in the space Up (0, T ) = m,k where (Y, Z) p = 0 (Y, Z)Sm [0, T ]×0 (0, T ) : (Y, Z) m×k <, a > 1 , sup epVs |Ys |p + E 0 e |Zs |2 p/2 BSVIs WITH LOCALLY BOUNDED GENERATORS 5. Appendix In this section we first present some useful and general estimates on 0 (Y, Z) Sm [0, T ] × 0 m×k (0, T ) satisfying an identity of type Yt = YT + dKs - Zs dBs , t [0, T ] , P-a.s., 0 where K Sm [0, T ] and K· () BV ([0, T ] ; Rm ) P-a.s., . The following results and their proofs are given in the monograph of s Pardoux, Ra¸canu [15], Annex Assume there exist D, R, N progressively measurable increasing continuous stochastic processes with D0 = R0 = N0 = 0, V progressively measurable bounded-variation continuous stochastic process with V0 = 0, a, p > 1, such that, as signed measures on [0, T ] , (31) dDt + Yt , dKt 1p2 dRt + |Yt |dNt + |Yt |2 dVt + np |Zt |2 dt, 2a . where np = (p - 1) 1. Let Y eV [t,T ] = sup Ys eVs and Y eV s[t,T ] T Y eV [0,T ] Proposition 11. Assume (31) and E Ye V p p/2 1p2 dRs e dNs Vs < . Then there exists a positive constant C , depending only of a, p, such that, P-a.s., for all t [0, T ] : EFt (32) sup |eVs Ys |p + s[t,T ] T e d p/2 e |Zs |2 p/2 + EFt epVs |Ys |p-2 1Ys =0 d + eVT YT p T epVs |Ys |p-2 1Ys =0 |Zs |2 p/2 T C EFt e 1p2 dRs eVs dNs ¸ L. MATICIUC, A. RASCANU and A. ZALINESCU In particular for all t [0, T ] : |Yt |p C EFt p p |YT |p + 1p2 RT + NT e p (V· -Vt )+ [t,T ] , P-a.s. Moreover if there exists a constant b 0 such that for all t [0, T ] : e VT -V 1/2 YT + 2(Vs -Vt ) 1p2 dRs e(Vs -Vt ) dNs b, a.s. then for all t [0, T ] : (33) |Yt |p + EFt p/2 e2(Vs -Vt ) |Zs |2 bp C , P-a.s. The following results provides a criterion for passing to the limit in Stieltjes integral (for the proofs we refer the reader to [15], Chapter I). Proposition 12. Let Y, K, Y n , K n be C ([0, T ] ; Rm )-valued random variables, n N. Assume (i) p > 0 such that sup E K n nN Kn - K p T (ii) ( Y n - Y T + - T ) - - 0, as n , n-Y n i.e. > 0, P {( Y T + K - K T ) > } 0, as n . Then, for all 0 s t T : t s n Yrn , dKr - - - p T prob. t prob. Yr , dKr , as n , s p T and moreover, E K lim inf n+ E K n Corollary 13. Lehe assumptions of Proposition 12 be satisfied. If A : Rm Rm is a (multivalued) maximal monotone operator then the following implication hol n dKt A (Ytn ) dt on [0, T ] , a.s. dKt A (Yt ) dt on [0, T ] , a.s. In particular if : Rd ] - , +] is a proper convex l.s. function then n dKt (Ytn ) dt on [0, T ] , a.s. dKt (Yt ) dt on [0, T ] , a.s. BSVIs WITH LOCALLY BOUNDED GENERATORS

Journal

Annals of the Alexandru Ioan Cuza University - Mathematics – de Gruyter

Published: Nov 24, 2014

There are no references for this article.